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Cotype for p-Banach spaces. (English) Zbl 0606.46011
Let $$X$$ be a real vector space, $$0<p\leq 1$$ and a p-convex norm on $$X$$, i.e. a function $$\| \cdot \|: X\to R_+$$ with the properties:
i) $$\| x\| >0$$, for $$x\neq 0$$
ii) $$\| ax\| =| a| \| x\|$$, $$a\in {\mathbb{R}}$$ and $$x\in X$$
iii) $$\| x+y\|^ p\leq \| x\|^ p+\| y\|^ p$$, $$x,y\in X.$$
Then $$(X,\| \cdot \|)$$ is a p-Banach space. And a p-Banach space $$(X,\| \cdot \|)$$ is of cotype $$q$$, $$0<q\leq +\infty$$, if there exists a constant $$C>0:$$ $$(\sum^{n}_{1}\| x_ i\|^ q)^{1/q}\leq C\int^{1}_{0}\| \sum^{n}_{1}r_ i(t)x_ i\| dt$$ for all $$n\in {\mathbb{N}}$$ and $$x_ i\in X$$, where $$\{r_ i(t)\}^ n_ 1$$ are the Rademacher functions on $$[0,1]$$.
Now in this paper the theorem of Maurey and Pisier for the cotype in p- Banach spaces is proved. Precisely it is proved that for $$(X,\| \cdot \|)$$ a real p-Banach space and $$q(X)=\inf \{q: X$$ is of $$q$$-Rademacher cotype$$\}\ell^{q(X)}$$ is finitely representable in $$X$$, when $$q(X)$$ is real and, for each $$\epsilon >0$$, $$c_ 0$$ is $$(2^{1/p-1}+\epsilon)$$ finitely representable in X, when $$q(X)=\infty$$.
Reviewer: K.Stathakopoulos
##### MSC:
 46B20 Geometry and structure of normed linear spaces 46B25 Classical Banach spaces in the general theory 46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
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##### References:
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